A system theoretic formulation of NMR experiments


A detailed system theoretic description is given of NMR experiments including relaxation effects. The approach is based on an exact and analytical solution to the master equation. It is shown that NMR experiments can be described in the framework of bilinear time-invariant systems. This description is used to derive closed-form expressions for the spectrum of one- and two-dimensional experiments. The simulations show that the approach accounts for the frequency dependence of a pulse, distinguishes between soft and hard pulses and also explains artifacts such as axial peaks.

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  1. [1]

    T. Kailath,Linear Systems (Prentice Hall, 1980).

  2. [2]

    M. Aoki,State Space Modelling of Time Series (Springer, 1987).

  3. [3]

    L. Ljung, System Identification (Prentice Hall, 1987).

  4. [4]

    R.R. Ernst, G. Bodenhausen and A. Wokaun,Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford University Press, 1987).

  5. [5]

    D.R. Brillinger and R. Kaiser, Fourier and likelihood analysis in NMR spectroscopy, Report No. 287, Department of Statistics, University of California (1981).

  6. [6]

    D.R. Brillinger, Some statistical aspects of NMR spectroscopy, Actas Del 2 Congreso Latinoamericano De Probabilidad Y Estadistica Matematica (1985).

  7. [7]

    R.A. Horn and C.R. Johnson,Topics in Matrix Analysis (Cambridge University Press, 1991).

  8. [8]

    W.J. Rugh,Nonlinear System Theory (The Johns Hopkins University Press, 1981).

  9. [9]

    R.K. Miller and A.N. Michel,Ordinary Differential Equations (Academic Press, 1982).

  10. [10]

    C.P. Slichter,Principles of Magnetic Resonance, 3rd ed. (Springer, 1992).

  11. [11]

    F.R.Gantmacher,The Theory of Matrices (Chelsea,1959).

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Ober, R.J., Ward, E.S. A system theoretic formulation of NMR experiments. J Math Chem 20, 47–65 (1996). https://doi.org/10.1007/BF01165155

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  • Physical Chemistry
  • Frequency Dependence
  • Master Equation
  • Theoretic Description
  • Relaxation Effect